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On positive functions with positive fourier transforms

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Fig. 1. Mixture of H 0 , H 4 and H 8 . White, no real positive ρ root; black, 4 roots;
Fig. 3. Left: Same as Fig. 2, with γ = 2π/15. Right: γ = π/6. See how the no root triangle shrinks.
Fig. 4. Left: γ = π/5; Right: γ = 7π/30. Absence of no root domain.
Fig. 6. Mixture of H 0 , H 2 and H 4 . White domain, 0 root. Grey, 1 root; black, 2 roots
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